The or rather one of the Frequentist interpretations of probability claims that the statement "The next coin toss )that is executed under conditions X) has a probability of 90% to land heads" simply means: "The next coin toss belongs to an infinite sequence of coin tosses that are executed under conditions X, and which have the limiting relative frequency 90%." Let (x1,x2, .... ) be the infinite sequence of coin tosses under condition X.

My question now is why this should give me any confidence that the next coin that I am interested in (and that is executed under conditions X) will land heads. It seems to me that the 90% are only relevant to my single case, if I make the additional assumption that the next coin toss that I am considering is somehow randomly chosen from this infinite sequence. Or alternatively I have symmetric evidence that the next coin toss be x1 or x2 or x3 or ... .

However I have never read anything similar to that from a defendant of the frequentist notion of probability

My real question is therefore whether:

  1. All that (Probability-)Frequentists mean when they claim: "The next coin toss (that is executed under conditions X) has a probability of 90% to land heads" is really: "The next coin toss belongs to an infinite sequence of coin tosses that are executed under conditions X, and which have the limiting relative frequency 90%." or
  2. Whether they actually mean something different and I just missunderstand them

1 Answer 1


It's useful to remember that frequentism was rooted in Machian or positivist radical empiricism. On this radical empiricist picture, all we can talk about are series of observations; we can't observe anything like "real chances" or stochastic causal connections, and so we should avoid that kind of language.

So, yes, #1, that really is all the most rigorous nineteenth century frequentists like Karl Pearson mean by probability.

Later frequentists were less rigorous empiricists, however. For example, Fisher was an indeterminist, and sometimes used stochastic causal connection language. The distribution of coin flip results in X can be described completely by an unobserved parameter rho (which you can interpret as the probability that any one coin flip is heads). For Fisher, rho has a true value, and rho stochastically causes the distribution of coin flip values that we actually observe. By virtue of this causal connection, we can use the percentage of observed flips that are heads to estimate the value of rho. For Karl Pearson, by contrast, this percentage is just a convenient way to summarize the data; since we can't actually observe rho, it's (roughly) meaningless to talk about its "true" value.

So #2 is true for an indeterminist frequentist like Fisher. There are "real chances" and stochastic causation, and we can estimate the true values of unobserved stochastic causes using the properties of long-run tendencies of sequences of observations.

  • What does it mean that something is stochastically caused? I think I am so biased by the idea of a (in the non-quantum world) deterministic universe that it does not make any sense to me at all to talk about "stochastic causation". Different outcomes arise due to a difference in the circumstances, so the parameters you mention are only useful because we lack some information about the situation in which the coin is tossed. They are a useful way to model situations not because the true parameter exists, but because we are incapable to model it in a deterministic way.
    – Sebastian
    Commented Mar 10, 2018 at 22:17
  • Two points. First, a stochastic causation or "real chances" view rejects determinism. So, yes, if you're assuming determinism, the idea of stochastic causation won't make much sense. The idea of stochastic causation was in a sense post-Newtonian or post-Laplacian, though statistical mechanics influenced the view well before quantum mechanics. For example, Peirce had a "real chances" view in the mid-nineteenth century, IIRC explicitly inspired by Maxwell.
    – Dan Hicks
    Commented Mar 11, 2018 at 17:04
  • Second, thinking about probability in terms of our epistemic/doxastic state comes from Bruno de Finetti. De Finetti's interpretation of probability is generally what people mean when they talk about "Bayesianism." IIRC de Finetti was the kind of radical empiricist who rejected both determinism and indeterminism — roughly, all we can talk about is our observations and our subjective degrees of belief.
    – Dan Hicks
    Commented Mar 11, 2018 at 17:11
  • So I guess this view of stochastic causation is similar to the propensity view on probability? I wonder which further conditions someone like Peirce had on the data to argue that it is caused stochastically. Assume that we get the infinite sequence: HTHTHTHTHTHT ... . The long run frequency of heads is 0.5. However there is another obvious model that predicts better which result will be obtained. There must be some kind of randomness in the sequence for a stochastic causation to be plausible. However what is/ seems random depends on our subjective judgment...
    – Sebastian
    Commented Mar 11, 2018 at 17:28

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